NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.0% → 98.9%

Time: 8.1s

Alternatives: 10

Speedup: 1.8×

• 38.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.99999999999999997e77
  2. eps = -2e-231

Specification

Math

\[\begin{array}{l} \\ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

C

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0;
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

C

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0;
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-23}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 4e-23)
   (/ (+ (exp (* x (+ eps -1.0))) (exp (* x (- eps)))) 2.0)
   (/ (+ (exp (- (* x eps) x)) (exp (- x))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 4e-23) {
		tmp = (exp((x * (eps + -1.0))) + exp((x * -eps))) / 2.0;
	} else {
		tmp = (exp(((x * eps) - x)) + exp(-x)) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 4d-23) then
        tmp = (exp((x * (eps + (-1.0d0)))) + exp((x * -eps))) / 2.0d0
    else
        tmp = (exp(((x * eps) - x)) + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 4e-23) {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp((x * -eps))) / 2.0;
	} else {
		tmp = (Math.exp(((x * eps) - x)) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 4e-23:
		tmp = (math.exp((x * (eps + -1.0))) + math.exp((x * -eps))) / 2.0
	else:
		tmp = (math.exp(((x * eps) - x)) + math.exp(-x)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 4e-23)
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(x * Float64(-eps)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(Float64(x * eps) - x)) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 4e-23)
		tmp = (exp((x * (eps + -1.0))) + exp((x * -eps))) / 2.0;
	else
		tmp = (exp(((x * eps) - x)) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 4e-23], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999984e-23

   1. Initial program 67.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 67.8%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 67.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 67.8%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 67.8%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 97.9%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 97.9%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 97.9%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 97.9%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 97.9%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 97.9%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 97.9%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 97.9%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 97.9%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 97.9%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around inf 97.9%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{\varepsilon \cdot x}}\right)}{2} \]

   if 3.99999999999999984e-23 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 100.0%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around 0 73.5%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x}}\right)}{2} \]

   8. Taylor expanded in x around inf 73.5%

      \[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot x - x} + e^{-x}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-23}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + e^{-x}}{2}\\ \end{array} \]

Alternative 2: 89.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\varepsilon \leq 3.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;\varepsilon \leq 7.2 \cdot 10^{+235} \lor \neg \left(\varepsilon \leq 2 \cdot 10^{+270}\right):\\ \;\;\;\;\frac{t_0 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \left(2 + x \cdot \left(\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps 3.4e-20)
     (/ (/ 2.0 (exp x)) 2.0)
     (if (or (<= eps 7.2e+235) (not (<= eps 2e+270)))
       (/ (+ t_0 (exp (* x eps))) 2.0)
       (/ (* t_0 (+ 2.0 (* x (+ eps (* x (* eps (* eps 0.5))))))) 2.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (eps <= 3.4e-20) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else if ((eps <= 7.2e+235) || !(eps <= 2e+270)) {
		tmp = (t_0 + exp((x * eps))) / 2.0;
	} else {
		tmp = (t_0 * (2.0 + (x * (eps + (x * (eps * (eps * 0.5))))))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps <= 3.4d-20) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else if ((eps <= 7.2d+235) .or. (.not. (eps <= 2d+270))) then
        tmp = (t_0 + exp((x * eps))) / 2.0d0
    else
        tmp = (t_0 * (2.0d0 + (x * (eps + (x * (eps * (eps * 0.5d0))))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps <= 3.4e-20) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else if ((eps <= 7.2e+235) || !(eps <= 2e+270)) {
		tmp = (t_0 + Math.exp((x * eps))) / 2.0;
	} else {
		tmp = (t_0 * (2.0 + (x * (eps + (x * (eps * (eps * 0.5))))))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = math.exp(-x)
	tmp = 0
	if eps <= 3.4e-20:
		tmp = (2.0 / math.exp(x)) / 2.0
	elif (eps <= 7.2e+235) or not (eps <= 2e+270):
		tmp = (t_0 + math.exp((x * eps))) / 2.0
	else:
		tmp = (t_0 * (2.0 + (x * (eps + (x * (eps * (eps * 0.5))))))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps <= 3.4e-20)
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	elseif ((eps <= 7.2e+235) || !(eps <= 2e+270))
		tmp = Float64(Float64(t_0 + exp(Float64(x * eps))) / 2.0);
	else
		tmp = Float64(Float64(t_0 * Float64(2.0 + Float64(x * Float64(eps + Float64(x * Float64(eps * Float64(eps * 0.5))))))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps <= 3.4e-20)
		tmp = (2.0 / exp(x)) / 2.0;
	elseif ((eps <= 7.2e+235) || ~((eps <= 2e+270)))
		tmp = (t_0 + exp((x * eps))) / 2.0;
	else
		tmp = (t_0 * (2.0 + (x * (eps + (x * (eps * (eps * 0.5))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps, 3.4e-20], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[eps, 7.2e+235], N[Not[LessEqual[eps, 2e+270]], $MachinePrecision]], N[(N[(t$95$0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 * N[(2.0 + N[(x * N[(eps + N[(x * N[(eps * N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 3.3999999999999997e-20

   1. Initial program 69.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 69.2%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 69.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 69.2%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 98.0%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 98.0%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 98.0%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 98.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 98.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 98.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 98.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 98.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 98.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 98.0%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around 0 80.7%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x}}\right)}{2} \]

   8. Taylor expanded in eps around 0 75.0%

      \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]
   9. Step-by-step derivation

      1. exp-neg 75.0%

         \[\leadsto \frac{2 \cdot \color{blue}{\frac{1}{e^{x}}}}{2} \]

      2. associate-*r/ 75.0%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{e^{x}}}}{2} \]

      3. metadata-eval 75.0%

         \[\leadsto \frac{\frac{\color{blue}{2}}{e^{x}}}{2} \]

   10. Simplified 75.0%

       \[\leadsto \frac{\color{blue}{\frac{2}{e^{x}}}}{2} \]

   if 3.3999999999999997e-20 < eps < 7.19999999999999971e235 or 2.0000000000000001e270 < eps

   1. Initial program 98.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 98.6%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 98.6%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 98.6%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 100.0%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around 0 86.4%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x}}\right)}{2} \]

   8. Taylor expanded in x around inf 86.4%

      \[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot x - x} + e^{-x}}}{2} \]

   9. Taylor expanded in eps around inf 86.4%

      \[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot x}} + e^{-x}}{2} \]
   10. Step-by-step derivation

       1. *-commutative 86.4%

          \[\leadsto \frac{e^{\color{blue}{x \cdot \varepsilon}} + e^{-x}}{2} \]

   11. Simplified 86.4%

       \[\leadsto \frac{e^{\color{blue}{x \cdot \varepsilon}} + e^{-x}}{2} \]

   if 7.19999999999999971e235 < eps < 2.0000000000000001e270

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 100.0%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around 0 46.2%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x}}\right)}{2} \]

   8. Taylor expanded in x around inf 46.2%

      \[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot x - x} + e^{-x}}}{2} \]

   9. Taylor expanded in eps around 0 50.0%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(e^{-x} \cdot x\right) + \left(0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right) + 2 \cdot e^{-x}\right)}}{2} \]
   10. Step-by-step derivation

       1. associate-+r+ 50.0%

          \[\leadsto \frac{\color{blue}{\left(\varepsilon \cdot \left(e^{-x} \cdot x\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right) + 2 \cdot e^{-x}}}{2} \]

       2. +-commutative 50.0%

          \[\leadsto \frac{\color{blue}{2 \cdot e^{-x} + \left(\varepsilon \cdot \left(e^{-x} \cdot x\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right)}}{2} \]

       3. *-commutative 50.0%

          \[\leadsto \frac{\color{blue}{e^{-x} \cdot 2} + \left(\varepsilon \cdot \left(e^{-x} \cdot x\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right)}{2} \]

       4. *-commutative 50.0%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\varepsilon \cdot \color{blue}{\left(x \cdot e^{-x}\right)} + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right)}{2} \]

       5. associate-*r* 50.0%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\color{blue}{\left(\varepsilon \cdot x\right) \cdot e^{-x}} + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right)}{2} \]

       6. associate-*r* 50.0%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\left(\varepsilon \cdot x\right) \cdot e^{-x} + \color{blue}{\left(0.5 \cdot {\varepsilon}^{2}\right) \cdot \left(e^{-x} \cdot {x}^{2}\right)}\right)}{2} \]

       7. *-commutative 50.0%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\left(\varepsilon \cdot x\right) \cdot e^{-x} + \left(0.5 \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot e^{-x}\right)}\right)}{2} \]

       8. associate-*r* 50.0%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\left(\varepsilon \cdot x\right) \cdot e^{-x} + \color{blue}{\left(\left(0.5 \cdot {\varepsilon}^{2}\right) \cdot {x}^{2}\right) \cdot e^{-x}}\right)}{2} \]

       9. distribute-rgt-out 50.0%

          \[\leadsto \frac{e^{-x} \cdot 2 + \color{blue}{e^{-x} \cdot \left(\varepsilon \cdot x + \left(0.5 \cdot {\varepsilon}^{2}\right) \cdot {x}^{2}\right)}}{2} \]

       10. distribute-lft-out 50.0%

           \[\leadsto \frac{\color{blue}{e^{-x} \cdot \left(2 + \left(\varepsilon \cdot x + \left(0.5 \cdot {\varepsilon}^{2}\right) \cdot {x}^{2}\right)\right)}}{2} \]

       11. unpow2 50.0%

           \[\leadsto \frac{e^{-x} \cdot \left(2 + \left(\varepsilon \cdot x + \left(0.5 \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{2} \]

   11. Simplified 67.7%

       \[\leadsto \frac{\color{blue}{e^{-x} \cdot \left(2 + x \cdot \left(\varepsilon + \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right) \cdot x\right)\right)}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;\varepsilon \leq 7.2 \cdot 10^{+235} \lor \neg \left(\varepsilon \leq 2 \cdot 10^{+270}\right):\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot \left(\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right)\right)\right)}{2}\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (/ (+ (exp (* x (+ eps -1.0))) (exp (* x (- -1.0 eps)))) 2.0))

C

eps = abs(eps);
double code(double x, double eps) {
	return (exp((x * (eps + -1.0))) + exp((x * (-1.0 - eps)))) / 2.0;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (exp((x * (eps + (-1.0d0)))) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	return (Math.exp((x * (eps + -1.0))) + Math.exp((x * (-1.0 - eps)))) / 2.0;
}

Python

eps = abs(eps)
def code(x, eps):
	return (math.exp((x * (eps + -1.0))) + math.exp((x * (-1.0 - eps)))) / 2.0

Julia

eps = abs(eps)
function code(x, eps)
	return Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0)
end

MATLAB

eps = abs(eps)
function tmp = code(x, eps)
	tmp = (exp((x * (eps + -1.0))) + exp((x * (-1.0 - eps)))) / 2.0;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 77.6%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Step-by-step derivation

   1. div-sub 77.6%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   2. +-rgt-identity 77.6%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. div-sub 77.6%

      \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

3. Simplified 77.6%

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

4. Taylor expanded in eps around inf 98.5%

   \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
5. Step-by-step derivation

   1. mul-1-neg 98.5%

      \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

   2. *-commutative 98.5%

      \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

   3. mul-1-neg 98.5%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

   4. exp-prod 98.5%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

   5. +-commutative 98.5%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

   6. *-lft-identity 98.5%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

   7. metadata-eval 98.5%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

   8. cancel-sign-sub-inv 98.5%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

   9. exp-prod 98.5%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

   10. mul-1-neg 98.5%

       \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

   11. *-commutative 98.5%

       \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

   12. cancel-sign-sub-inv 98.5%

       \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

   13. metadata-eval 98.5%

       \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

   14. *-lft-identity 98.5%

       \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

   15. +-commutative 98.5%

       \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

6. Simplified 98.5%

   \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

7. Final simplification 98.5%

   \[\leadsto \frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

Alternative 4: 95.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-231}:\\ \;\;\;\;\frac{t_0 \cdot \left(2 + x \cdot \left(\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + t_0}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1.6e-231)
     (/ (* t_0 (+ 2.0 (* x (+ eps (* x (* eps (* eps 0.5))))))) 2.0)
     (/ (+ (exp (- (* x eps) x)) t_0) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1.6e-231) {
		tmp = (t_0 * (2.0 + (x * (eps + (x * (eps * (eps * 0.5))))))) / 2.0;
	} else {
		tmp = (exp(((x * eps) - x)) + t_0) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-1.6d-231)) then
        tmp = (t_0 * (2.0d0 + (x * (eps + (x * (eps * (eps * 0.5d0))))))) / 2.0d0
    else
        tmp = (exp(((x * eps) - x)) + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (x <= -1.6e-231) {
		tmp = (t_0 * (2.0 + (x * (eps + (x * (eps * (eps * 0.5))))))) / 2.0;
	} else {
		tmp = (Math.exp(((x * eps) - x)) + t_0) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -1.6e-231:
		tmp = (t_0 * (2.0 + (x * (eps + (x * (eps * (eps * 0.5))))))) / 2.0
	else:
		tmp = (math.exp(((x * eps) - x)) + t_0) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1.6e-231)
		tmp = Float64(Float64(t_0 * Float64(2.0 + Float64(x * Float64(eps + Float64(x * Float64(eps * Float64(eps * 0.5))))))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(Float64(x * eps) - x)) + t_0) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = exp(-x);
	tmp = 0.0;
	if (x <= -1.6e-231)
		tmp = (t_0 * (2.0 + (x * (eps + (x * (eps * (eps * 0.5))))))) / 2.0;
	else
		tmp = (exp(((x * eps) - x)) + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1.6e-231], N[(N[(t$95$0 * N[(2.0 + N[(x * N[(eps + N[(x * N[(eps * N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.60000000000000004e-231

   1. Initial program 73.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 73.5%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 73.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 73.5%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 96.2%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 96.2%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 96.2%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 96.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 96.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 96.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 96.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 96.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 96.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 96.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 96.2%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 96.2%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 96.2%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 96.2%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 96.2%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 96.2%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 96.2%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around 0 84.5%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x}}\right)}{2} \]

   8. Taylor expanded in x around inf 84.5%

      \[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot x - x} + e^{-x}}}{2} \]

   9. Taylor expanded in eps around 0 66.2%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(e^{-x} \cdot x\right) + \left(0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right) + 2 \cdot e^{-x}\right)}}{2} \]
   10. Step-by-step derivation

       1. associate-+r+ 66.2%

          \[\leadsto \frac{\color{blue}{\left(\varepsilon \cdot \left(e^{-x} \cdot x\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right) + 2 \cdot e^{-x}}}{2} \]

       2. +-commutative 66.2%

          \[\leadsto \frac{\color{blue}{2 \cdot e^{-x} + \left(\varepsilon \cdot \left(e^{-x} \cdot x\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right)}}{2} \]

       3. *-commutative 66.2%

          \[\leadsto \frac{\color{blue}{e^{-x} \cdot 2} + \left(\varepsilon \cdot \left(e^{-x} \cdot x\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right)}{2} \]

       4. *-commutative 66.2%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\varepsilon \cdot \color{blue}{\left(x \cdot e^{-x}\right)} + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right)}{2} \]

       5. associate-*r* 66.2%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\color{blue}{\left(\varepsilon \cdot x\right) \cdot e^{-x}} + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right)}{2} \]

       6. associate-*r* 66.2%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\left(\varepsilon \cdot x\right) \cdot e^{-x} + \color{blue}{\left(0.5 \cdot {\varepsilon}^{2}\right) \cdot \left(e^{-x} \cdot {x}^{2}\right)}\right)}{2} \]

       7. *-commutative 66.2%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\left(\varepsilon \cdot x\right) \cdot e^{-x} + \left(0.5 \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot e^{-x}\right)}\right)}{2} \]

       8. associate-*r* 66.2%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\left(\varepsilon \cdot x\right) \cdot e^{-x} + \color{blue}{\left(\left(0.5 \cdot {\varepsilon}^{2}\right) \cdot {x}^{2}\right) \cdot e^{-x}}\right)}{2} \]

       9. distribute-rgt-out 74.3%

          \[\leadsto \frac{e^{-x} \cdot 2 + \color{blue}{e^{-x} \cdot \left(\varepsilon \cdot x + \left(0.5 \cdot {\varepsilon}^{2}\right) \cdot {x}^{2}\right)}}{2} \]

       10. distribute-lft-out 74.3%

           \[\leadsto \frac{\color{blue}{e^{-x} \cdot \left(2 + \left(\varepsilon \cdot x + \left(0.5 \cdot {\varepsilon}^{2}\right) \cdot {x}^{2}\right)\right)}}{2} \]

       11. unpow2 74.3%

           \[\leadsto \frac{e^{-x} \cdot \left(2 + \left(\varepsilon \cdot x + \left(0.5 \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{2} \]

   11. Simplified 89.5%

       \[\leadsto \frac{\color{blue}{e^{-x} \cdot \left(2 + x \cdot \left(\varepsilon + \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right) \cdot x\right)\right)}}{2} \]

   if -1.60000000000000004e-231 < x

   1. Initial program 80.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 80.3%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 80.3%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 80.3%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 80.3%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 100.0%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around 0 79.4%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x}}\right)}{2} \]

   8. Taylor expanded in x around inf 79.4%

      \[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot x - x} + e^{-x}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-231}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot \left(\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + e^{-x}}{2}\\ \end{array} \]

Alternative 5: 77.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \frac{\frac{2}{e^{x}}}{2}\\ \mathbf{if}\;\varepsilon \leq 3700000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{2 + x \cdot \left(x + -2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 9.2 \cdot 10^{+174}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot \left(\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 (exp x)) 2.0)))
   (if (<= eps 3700000000000.0)
     t_0
     (if (<= eps 1.1e+83)
       (/ (+ 2.0 (* x (+ x -2.0))) 2.0)
       (if (<= eps 9.2e+174)
         t_0
         (/
          (* (exp (- x)) (+ 2.0 (* x (+ eps (* x (* eps (* eps 0.5)))))))
          2.0))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (2.0 / exp(x)) / 2.0;
	double tmp;
	if (eps <= 3700000000000.0) {
		tmp = t_0;
	} else if (eps <= 1.1e+83) {
		tmp = (2.0 + (x * (x + -2.0))) / 2.0;
	} else if (eps <= 9.2e+174) {
		tmp = t_0;
	} else {
		tmp = (exp(-x) * (2.0 + (x * (eps + (x * (eps * (eps * 0.5))))))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / exp(x)) / 2.0d0
    if (eps <= 3700000000000.0d0) then
        tmp = t_0
    else if (eps <= 1.1d+83) then
        tmp = (2.0d0 + (x * (x + (-2.0d0)))) / 2.0d0
    else if (eps <= 9.2d+174) then
        tmp = t_0
    else
        tmp = (exp(-x) * (2.0d0 + (x * (eps + (x * (eps * (eps * 0.5d0))))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = (2.0 / Math.exp(x)) / 2.0;
	double tmp;
	if (eps <= 3700000000000.0) {
		tmp = t_0;
	} else if (eps <= 1.1e+83) {
		tmp = (2.0 + (x * (x + -2.0))) / 2.0;
	} else if (eps <= 9.2e+174) {
		tmp = t_0;
	} else {
		tmp = (Math.exp(-x) * (2.0 + (x * (eps + (x * (eps * (eps * 0.5))))))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = (2.0 / math.exp(x)) / 2.0
	tmp = 0
	if eps <= 3700000000000.0:
		tmp = t_0
	elif eps <= 1.1e+83:
		tmp = (2.0 + (x * (x + -2.0))) / 2.0
	elif eps <= 9.2e+174:
		tmp = t_0
	else:
		tmp = (math.exp(-x) * (2.0 + (x * (eps + (x * (eps * (eps * 0.5))))))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(2.0 / exp(x)) / 2.0)
	tmp = 0.0
	if (eps <= 3700000000000.0)
		tmp = t_0;
	elseif (eps <= 1.1e+83)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(x + -2.0))) / 2.0);
	elseif (eps <= 9.2e+174)
		tmp = t_0;
	else
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * Float64(eps + Float64(x * Float64(eps * Float64(eps * 0.5))))))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = (2.0 / exp(x)) / 2.0;
	tmp = 0.0;
	if (eps <= 3700000000000.0)
		tmp = t_0;
	elseif (eps <= 1.1e+83)
		tmp = (2.0 + (x * (x + -2.0))) / 2.0;
	elseif (eps <= 9.2e+174)
		tmp = t_0;
	else
		tmp = (exp(-x) * (2.0 + (x * (eps + (x * (eps * (eps * 0.5))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[eps, 3700000000000.0], t$95$0, If[LessEqual[eps, 1.1e+83], N[(N[(2.0 + N[(x * N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 9.2e+174], t$95$0, N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * N[(eps + N[(x * N[(eps * N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 3.7e12 or 1.09999999999999999e83 < eps < 9.1999999999999991e174

   1. Initial program 72.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 72.5%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 72.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 72.5%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 72.5%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 98.2%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 98.2%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 98.2%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 98.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 98.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 98.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 98.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 98.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 98.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 98.2%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 98.2%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 98.2%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 98.2%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 98.2%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 98.2%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 98.2%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 98.2%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around 0 81.2%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x}}\right)}{2} \]

   8. Taylor expanded in eps around 0 72.9%

      \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]
   9. Step-by-step derivation

      1. exp-neg 72.9%

         \[\leadsto \frac{2 \cdot \color{blue}{\frac{1}{e^{x}}}}{2} \]

      2. associate-*r/ 72.9%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{e^{x}}}}{2} \]

      3. metadata-eval 72.9%

         \[\leadsto \frac{\frac{\color{blue}{2}}{e^{x}}}{2} \]

   10. Simplified 72.9%

       \[\leadsto \frac{\color{blue}{\frac{2}{e^{x}}}}{2} \]

   if 3.7e12 < eps < 1.09999999999999999e83

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 100.0%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around 0 100.0%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x}}\right)}{2} \]

   8. Taylor expanded in eps around 0 67.3%

      \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]
   9. Step-by-step derivation

      1. exp-neg 67.3%

         \[\leadsto \frac{2 \cdot \color{blue}{\frac{1}{e^{x}}}}{2} \]

      2. associate-*r/ 67.3%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{e^{x}}}}{2} \]

      3. metadata-eval 67.3%

         \[\leadsto \frac{\frac{\color{blue}{2}}{e^{x}}}{2} \]

   10. Simplified 67.3%

       \[\leadsto \frac{\color{blue}{\frac{2}{e^{x}}}}{2} \]

   11. Taylor expanded in x around 0 82.0%

       \[\leadsto \frac{\color{blue}{2 + \left(-2 \cdot x + {x}^{2}\right)}}{2} \]
   12. Step-by-step derivation

       1. unpow2 82.0%

          \[\leadsto \frac{2 + \left(-2 \cdot x + \color{blue}{x \cdot x}\right)}{2} \]

       2. distribute-rgt-out 82.0%

          \[\leadsto \frac{2 + \color{blue}{x \cdot \left(-2 + x\right)}}{2} \]

       3. +-commutative 82.0%

          \[\leadsto \frac{2 + x \cdot \color{blue}{\left(x + -2\right)}}{2} \]

   13. Simplified 82.0%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(x + -2\right)}}{2} \]

   if 9.1999999999999991e174 < eps

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 100.0%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around 0 68.6%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x}}\right)}{2} \]

   8. Taylor expanded in x around inf 68.6%

      \[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot x - x} + e^{-x}}}{2} \]

   9. Taylor expanded in eps around 0 29.6%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(e^{-x} \cdot x\right) + \left(0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right) + 2 \cdot e^{-x}\right)}}{2} \]
   10. Step-by-step derivation

       1. associate-+r+ 29.6%

          \[\leadsto \frac{\color{blue}{\left(\varepsilon \cdot \left(e^{-x} \cdot x\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right) + 2 \cdot e^{-x}}}{2} \]

       2. +-commutative 29.6%

          \[\leadsto \frac{\color{blue}{2 \cdot e^{-x} + \left(\varepsilon \cdot \left(e^{-x} \cdot x\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right)}}{2} \]

       3. *-commutative 29.6%

          \[\leadsto \frac{\color{blue}{e^{-x} \cdot 2} + \left(\varepsilon \cdot \left(e^{-x} \cdot x\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right)}{2} \]

       4. *-commutative 29.6%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\varepsilon \cdot \color{blue}{\left(x \cdot e^{-x}\right)} + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right)}{2} \]

       5. associate-*r* 29.6%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\color{blue}{\left(\varepsilon \cdot x\right) \cdot e^{-x}} + 0.5 \cdot \left({\varepsilon}^{2} \cdot \left(e^{-x} \cdot {x}^{2}\right)\right)\right)}{2} \]

       6. associate-*r* 29.6%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\left(\varepsilon \cdot x\right) \cdot e^{-x} + \color{blue}{\left(0.5 \cdot {\varepsilon}^{2}\right) \cdot \left(e^{-x} \cdot {x}^{2}\right)}\right)}{2} \]

       7. *-commutative 29.6%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\left(\varepsilon \cdot x\right) \cdot e^{-x} + \left(0.5 \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot e^{-x}\right)}\right)}{2} \]

       8. associate-*r* 29.6%

          \[\leadsto \frac{e^{-x} \cdot 2 + \left(\left(\varepsilon \cdot x\right) \cdot e^{-x} + \color{blue}{\left(\left(0.5 \cdot {\varepsilon}^{2}\right) \cdot {x}^{2}\right) \cdot e^{-x}}\right)}{2} \]

       9. distribute-rgt-out 33.3%

          \[\leadsto \frac{e^{-x} \cdot 2 + \color{blue}{e^{-x} \cdot \left(\varepsilon \cdot x + \left(0.5 \cdot {\varepsilon}^{2}\right) \cdot {x}^{2}\right)}}{2} \]

       10. distribute-lft-out 33.3%

           \[\leadsto \frac{\color{blue}{e^{-x} \cdot \left(2 + \left(\varepsilon \cdot x + \left(0.5 \cdot {\varepsilon}^{2}\right) \cdot {x}^{2}\right)\right)}}{2} \]

       11. unpow2 33.3%

           \[\leadsto \frac{e^{-x} \cdot \left(2 + \left(\varepsilon \cdot x + \left(0.5 \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{2} \]

   11. Simplified 74.5%

       \[\leadsto \frac{\color{blue}{e^{-x} \cdot \left(2 + x \cdot \left(\varepsilon + \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right) \cdot x\right)\right)}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3700000000000:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{2 + x \cdot \left(x + -2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 9.2 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot \left(\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right)\right)\right)}{2}\\ \end{array} \]

Alternative 6: 70.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8200000000000 \lor \neg \left(\varepsilon \leq 1.85 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(x + -2\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps 8200000000000.0) (not (<= eps 1.85e+82)))
   (/ (/ 2.0 (exp x)) 2.0)
   (/ (+ 2.0 (* x (+ x -2.0))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if ((eps <= 8200000000000.0) || !(eps <= 1.85e+82)) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (x + -2.0))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= 8200000000000.0d0) .or. (.not. (eps <= 1.85d+82))) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * (x + (-2.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if ((eps <= 8200000000000.0) || !(eps <= 1.85e+82)) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (x + -2.0))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if (eps <= 8200000000000.0) or not (eps <= 1.85e+82):
		tmp = (2.0 / math.exp(x)) / 2.0
	else:
		tmp = (2.0 + (x * (x + -2.0))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if ((eps <= 8200000000000.0) || !(eps <= 1.85e+82))
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(x + -2.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= 8200000000000.0) || ~((eps <= 1.85e+82)))
		tmp = (2.0 / exp(x)) / 2.0;
	else
		tmp = (2.0 + (x * (x + -2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[Or[LessEqual[eps, 8200000000000.0], N[Not[LessEqual[eps, 1.85e+82]], $MachinePrecision]], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 8.2e12 or 1.8500000000000001e82 < eps

   1. Initial program 75.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 75.6%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 75.6%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 75.6%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 98.4%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 98.4%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 98.4%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 98.4%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 98.4%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 98.4%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 98.4%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 98.4%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 98.4%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 98.4%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 98.4%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 98.4%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 98.4%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 98.4%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 98.4%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 98.4%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 98.4%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around 0 79.7%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x}}\right)}{2} \]

   8. Taylor expanded in eps around 0 69.8%

      \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]
   9. Step-by-step derivation

      1. exp-neg 69.8%

         \[\leadsto \frac{2 \cdot \color{blue}{\frac{1}{e^{x}}}}{2} \]

      2. associate-*r/ 69.8%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{e^{x}}}}{2} \]

      3. metadata-eval 69.8%

         \[\leadsto \frac{\frac{\color{blue}{2}}{e^{x}}}{2} \]

   10. Simplified 69.8%

       \[\leadsto \frac{\color{blue}{\frac{2}{e^{x}}}}{2} \]

   if 8.2e12 < eps < 1.8500000000000001e82

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 100.0%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 100.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 100.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around 0 100.0%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x}}\right)}{2} \]

   8. Taylor expanded in eps around 0 67.3%

      \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]
   9. Step-by-step derivation

      1. exp-neg 67.3%

         \[\leadsto \frac{2 \cdot \color{blue}{\frac{1}{e^{x}}}}{2} \]

      2. associate-*r/ 67.3%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{e^{x}}}}{2} \]

      3. metadata-eval 67.3%

         \[\leadsto \frac{\frac{\color{blue}{2}}{e^{x}}}{2} \]

   10. Simplified 67.3%

       \[\leadsto \frac{\color{blue}{\frac{2}{e^{x}}}}{2} \]

   11. Taylor expanded in x around 0 82.0%

       \[\leadsto \frac{\color{blue}{2 + \left(-2 \cdot x + {x}^{2}\right)}}{2} \]
   12. Step-by-step derivation

       1. unpow2 82.0%

          \[\leadsto \frac{2 + \left(-2 \cdot x + \color{blue}{x \cdot x}\right)}{2} \]

       2. distribute-rgt-out 82.0%

          \[\leadsto \frac{2 + \color{blue}{x \cdot \left(-2 + x\right)}}{2} \]

       3. +-commutative 82.0%

          \[\leadsto \frac{2 + x \cdot \color{blue}{\left(x + -2\right)}}{2} \]

   13. Simplified 82.0%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(x + -2\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8200000000000 \lor \neg \left(\varepsilon \leq 1.85 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(x + -2\right)}{2}\\ \end{array} \]

Alternative 7: 63.5% accurate, 14.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1150 \lor \neg \left(x \leq 1.75 \cdot 10^{+260}\right) \land x \leq 8 \cdot 10^{+301}:\\ \;\;\;\;\frac{2 + x \cdot \left(x + -2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (or (<= x 1150.0) (and (not (<= x 1.75e+260)) (<= x 8e+301)))
   (/ (+ 2.0 (* x (+ x -2.0))) 2.0)
   0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if ((x <= 1150.0) || (!(x <= 1.75e+260) && (x <= 8e+301))) {
		tmp = (2.0 + (x * (x + -2.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= 1150.0d0) .or. (.not. (x <= 1.75d+260)) .and. (x <= 8d+301)) then
        tmp = (2.0d0 + (x * (x + (-2.0d0)))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if ((x <= 1150.0) || (!(x <= 1.75e+260) && (x <= 8e+301))) {
		tmp = (2.0 + (x * (x + -2.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if (x <= 1150.0) or (not (x <= 1.75e+260) and (x <= 8e+301)):
		tmp = (2.0 + (x * (x + -2.0))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if ((x <= 1150.0) || (!(x <= 1.75e+260) && (x <= 8e+301)))
		tmp = Float64(Float64(2.0 + Float64(x * Float64(x + -2.0))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= 1150.0) || (~((x <= 1.75e+260)) && (x <= 8e+301)))
		tmp = (2.0 + (x * (x + -2.0))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[Or[LessEqual[x, 1150.0], And[N[Not[LessEqual[x, 1.75e+260]], $MachinePrecision], LessEqual[x, 8e+301]]], N[(N[(2.0 + N[(x * N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1150 or 1.7499999999999999e260 < x < 8.00000000000000042e301

   1. Initial program 70.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 70.0%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 70.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 70.0%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 98.0%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 98.0%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 98.0%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 98.0%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 98.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 98.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 98.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 98.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 98.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 98.0%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 98.0%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around 0 83.3%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x}}\right)}{2} \]

   8. Taylor expanded in eps around 0 73.1%

      \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]
   9. Step-by-step derivation

      1. exp-neg 73.1%

         \[\leadsto \frac{2 \cdot \color{blue}{\frac{1}{e^{x}}}}{2} \]

      2. associate-*r/ 73.1%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{e^{x}}}}{2} \]

      3. metadata-eval 73.1%

         \[\leadsto \frac{\frac{\color{blue}{2}}{e^{x}}}{2} \]

   10. Simplified 73.1%

       \[\leadsto \frac{\color{blue}{\frac{2}{e^{x}}}}{2} \]

   11. Taylor expanded in x around 0 68.8%

       \[\leadsto \frac{\color{blue}{2 + \left(-2 \cdot x + {x}^{2}\right)}}{2} \]
   12. Step-by-step derivation

       1. unpow2 68.8%

          \[\leadsto \frac{2 + \left(-2 \cdot x + \color{blue}{x \cdot x}\right)}{2} \]

       2. distribute-rgt-out 68.8%

          \[\leadsto \frac{2 + \color{blue}{x \cdot \left(-2 + x\right)}}{2} \]

       3. +-commutative 68.8%

          \[\leadsto \frac{2 + x \cdot \color{blue}{\left(x + -2\right)}}{2} \]

   13. Simplified 68.8%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(x + -2\right)}}{2} \]

   if 1150 < x < 1.7499999999999999e260 or 8.00000000000000042e301 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{\varepsilon + -1}\right)}^{x}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]

   4. Taylor expanded in eps around 0 59.1%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]
   5. Step-by-step derivation

      1. div-sub 59.1%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      2. rec-exp 59.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2} \]

      3. mul-1-neg 59.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]

      4. +-inverses 59.1%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 59.1%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1150 \lor \neg \left(x \leq 1.75 \cdot 10^{+260}\right) \land x \leq 8 \cdot 10^{+301}:\\ \;\;\;\;\frac{2 + x \cdot \left(x + -2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 56.9% accurate, 25.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{2 + x \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 1.0) (/ (+ 2.0 (* x -2.0)) 2.0) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 1.0) {
		tmp = (2.0 + (x * -2.0)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (2.0d0 + (x * (-2.0d0))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.0) {
		tmp = (2.0 + (x * -2.0)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 1.0:
		tmp = (2.0 + (x * -2.0)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(2.0 + Float64(x * -2.0)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (2.0 + (x * -2.0)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 1.0], N[(N[(2.0 + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 68.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 68.2%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 68.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 68.2%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around inf 97.9%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 97.9%

         \[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      2. *-commutative 97.9%

         \[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]

      3. mul-1-neg 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]

      4. exp-prod 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]

      5. +-commutative 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      6. *-lft-identity 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}\right)}{2} \]

      7. metadata-eval 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}\right)}{2} \]

      8. cancel-sign-sub-inv 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}\right)}{2} \]

      9. exp-prod 97.9%

         \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}\right)}{2} \]

      10. mul-1-neg 97.9%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]

      11. *-commutative 97.9%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]

      12. cancel-sign-sub-inv 97.9%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)}}\right)}{2} \]

      13. metadata-eval 97.9%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{1} \cdot \varepsilon\right)}\right)}{2} \]

      14. *-lft-identity 97.9%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}\right)}{2} \]

      15. +-commutative 97.9%

          \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   6. Simplified 97.9%

      \[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}}{2} \]

   7. Taylor expanded in eps around 0 84.5%

      \[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\color{blue}{x}}\right)}{2} \]

   8. Taylor expanded in eps around 0 76.4%

      \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]
   9. Step-by-step derivation

      1. exp-neg 76.4%

         \[\leadsto \frac{2 \cdot \color{blue}{\frac{1}{e^{x}}}}{2} \]

      2. associate-*r/ 76.4%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{e^{x}}}}{2} \]

      3. metadata-eval 76.4%

         \[\leadsto \frac{\frac{\color{blue}{2}}{e^{x}}}{2} \]

   10. Simplified 76.4%

       \[\leadsto \frac{\color{blue}{\frac{2}{e^{x}}}}{2} \]

   11. Taylor expanded in x around 0 56.0%

       \[\leadsto \frac{\color{blue}{2 + -2 \cdot x}}{2} \]
   12. Step-by-step derivation

       1. *-commutative 56.0%

          \[\leadsto \frac{2 + \color{blue}{x \cdot -2}}{2} \]

   13. Simplified 56.0%

       \[\leadsto \frac{\color{blue}{2 + x \cdot -2}}{2} \]

   if 1 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{\varepsilon + -1}\right)}^{x}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]

   4. Taylor expanded in eps around 0 53.4%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]
   5. Step-by-step derivation

      1. div-sub 53.4%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      2. rec-exp 53.4%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2} \]

      3. mul-1-neg 53.4%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]

      4. +-inverses 53.4%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 53.4%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{2 + x \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 56.8% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1150:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 1150.0) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 1150.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1150.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 1150.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 1150.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 1150.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1150.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 1150.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1150

   1. Initial program 68.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. div-sub 68.4%

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

      2. +-rgt-identity 68.4%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 68.4%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in x around 0 55.2%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 1150 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{\varepsilon + -1}\right)}^{x}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]

   4. Taylor expanded in eps around 0 54.1%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]
   5. Step-by-step derivation

      1. div-sub 54.1%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      2. rec-exp 54.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2} \]

      3. mul-1-neg 54.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]

      4. +-inverses 54.1%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 54.1%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1150:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 16.5% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ 0 \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 0.0)

C

eps = abs(eps);
double code(double x, double eps) {
	return 0.0;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	return 0.0;
}

Python

eps = abs(eps)
def code(x, eps):
	return 0.0

Julia

eps = abs(eps)
function code(x, eps)
	return 0.0
end

MATLAB

eps = abs(eps)
function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := 0.0

Derivation

1. Initial program 77.6%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Step-by-step derivation

3. Simplified 64.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{\varepsilon + -1}\right)}^{x}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]

4. Taylor expanded in eps around 0 17.3%

   \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]
5. Step-by-step derivation

   1. div-sub 17.3%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

   2. rec-exp 17.3%

      \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2} \]

   3. mul-1-neg 17.3%

      \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]

   4. +-inverses 17.5%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

6. Simplified 17.5%

   \[\leadsto \frac{\color{blue}{0}}{2} \]

7. Final simplification 17.5%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023250 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))